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Math Help - finding equivalence classes

  1. #1
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    finding equivalence classes

    S is the set of all real #'s

    relation R on S is:

    R= ((a,b) : a,b are elements of S and a-b is an integer)

    R is reflexive, transitive and symmetric, making it an equivalence relation.

    how do i find the equivalence classes? i know there are infinitely many.

    thanks.
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by CSkyle View Post
    S is the set of all real #'s

    relation R on S is:

    R= ((a,b) : a,b are elements of S and a-b is an integer)

    R is reflexive, transitive and symmetric, making it an equivalence relation.

    how do i find the equivalence classes? i know there are infinitely many.

    thanks.
    Hint: we want a - b = k for some integer k. that means, a = k + b. what do you think forms the equivalence class of the element a?
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  3. #3
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    so, the equivalence classes for R consist of all sets (a,b) with a,b being elements of S, that have the form ((k+b),(a-k)) for some integer K ?
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  4. #4
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    Quote Originally Posted by CSkyle View Post
    so, the equivalence classes for R consist of all sets (a,b) with a,b being elements of S, that have the form ((k+b),(a-k)) for some integer K ?
    If a \in \mathbb{R}, [a] = a + \mathbb{Z}

    For example:
    [\pi] = \{\pi,\pi \pm 1,\pi \pm 2,......\}
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